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AUTOCORRELATION: WHAT HAPPENS IF THE ERROR TERMS ARE CORRELATED?

The reader may recall that there are generally three types of data that are available for empirical analysis: (1) cross section, (2) time series, and

(3) combination of cross section and time series, also known as pooled data. In developing the classical linear regression model (CLRM) in Part I we made several assumptions, which were discussed in Section 7.1. However, we noted that not all these assumptions would hold in every type of data. As a matter of fact, we saw in the previous chapter that the assumption of homoscedasticity, or equal error variance, may not be always tenable in cross-sectional data. In other words, cross-sectional data are often plagued by the problem of heteroscedasticity.

However, in cross-section studies, data are often collected on the basis of a random sample of cross-sectional units, such as households (in a consumption function analysis) or firms (in an investment study analysis) so that there is no prior reason to believe that the error term pertaining to one household or a firm is correlated with the error term of another household or firm. If by chance such a correlation is observed in cross-sectional units, it is called spatial autocorrelation, that is, correlation in space rather than over time. However, it is important to remember that, in cross-sectional analysis, the ordering of the data must have some logic, or economic interest, to make sense of any determination of whether (spatial) autocorrelation is present or not.

The situation, however, is likely to be very different if we are dealing with time series data, for the observations in such data follow a natural ordering over time so that successive observations are likely to exhibit intercorrela-tions, especially if the time interval between successive observations is

442 PART TWO: RELAXING THE ASSUMPTIONS OF THE CLASSICAL MODEL

short, such as a day, a week, or a month rather than a year. If you observe stock price indexes, such as the Dow Jones or S&P 500 over successive days, it is not unusual to find that these indexes move up or down for several days in succession. Obviously, in situations like this, the assumption of no auto, or serial, correlation in the error terms that underlies the CLRM will be violated.

In this chapter we take a critical look at this assumption with a view to answering the following questions:

1. What is the nature of autocorrelation?

2. What are the theoretical and practical consequences of autocorrelation?

3. Since the assumption of no autocorrelation relates to the unobserv-able disturbances ut, how does one know that there is autocorrelation in any given situation? Notice that we now use the subscript t to emphasize that we are dealing with time series data.

4. How does one remedy the problem of autocorrelation?

The reader will find this chapter in many ways similar to the preceding chapter on heteroscedasticity in that under both heteroscedasticity and autocorrelation the usual OLS estimators, although linear, unbiased, and asymptotically (i.e., in large samples) normally distributed,1 are no longer minimum variance among all linear unbiased estimators. In short, they are not efficient relative to other linear and unbiased estimators. Put differently, they may not be BLUE. As a result, the usual, t, F, and x2 may not be valid.

The term autocorrelation may be defined as "correlation between members of series of observations ordered in time [as in time series data] or space [as in cross-sectional data].''2 In the regression context, the classical linear regression model assumes that such autocorrelation does not exist in the disturbances ui . Symbolically,

Put simply, the classical model assumes that the disturbance term relating to any observation is not influenced by the disturbance term relating to any other observation. For example, if we are dealing with quarterly time series data involving the regression of output on labor and capital inputs and if,

'On this, see William H. Greene, Econometric Analysis, 4th ed., Prentice Hall, N.J., 2000, Chap. 11, and Paul A. Rudd, An Introduction to Classical Econometric Theory, Oxford University Press, 2000, Chap. 19.

2Maurice G. Kendall and William R. Buckland, A Dictionary of Statistical Terms, Hafner Publishing Company, New York, 1971, p. 8. 